Leave the center taps disconnected, since with floating center taps there'll be no chance of imbalance in the 12.6 volt windings causing damage anywhere.
Also, be sure that your variac drives the transformer array instead of the array driving the variac, since that way the array won't have to supply the variac's losses.
This answer explains that for an unloaded secondary, the natural phase relationship between primary voltage and secondary voltage is is zero degrees.
It therefore follows that if there is a secondary load current (due to a resistive load), the current in the primary due to that secondary resistive load must be 180 degrees out of phase with the secondary load current i.e. as current flows into the primary, current flows out from the secondary.
This of course is for an ideal transformer and a resistive load.
If you ignore the leakages and magnetization inductance of the transformer, and the load is reactive, then there will be a 90 degrees phase shift.
Bringing in leakage inductance and DC coil resistance will/can muddy the waters. Bringing in magnetization inductance muddies the water a bit more.
The low frequency transformer equivalent circuit is this: -
As you should be able to see, if you considered all the leakages, magnetization inductance and losses and then added a semi-reactive load, the phase angle is quite complex to calculate.
However I do-not know, in which-way Lenz's law acts in transformer;
because the law states the induced current will try to hinder the
cause.
Strictly speaking, it is voltage that is induced and any current that flows is subject to the that voltage, the load and the leakage inductance.
but when the secondary circuit of a step-up transformer turned On
(closed) , so-far I've know , the current in primary-coil goes-up
In normal usage, for a voltage transformer it is non-ideal to consider the secondary being short circuited. However, it makes no difference to the phase angle providing you obey the rules inherent to the model.
Best Answer
Using a centre-tapped primary usually means that you de-rate the transformer to half power.
Figure 1. Series and parallel connections for split-winding transformer. Image source: Electro-Dan. The linked article discusses torroidal transformers but the principle is the same for standard transformers.
Instead, the common approach due to its flexibility is to use a transformer with two 110 V primaries and two equal secondaries. The primaries can then be wired in series for 220 V operation and parallel for 110 V. Similarly, the secondaries can be wired in parallel for high current or series for higher voltage. The series connection point is, in effect, the secondary centre-tap.
Figure 2. For switching primary voltage by the user you would add a voltage selector switch. Source: Modulus Amplification.